Theorem ltexnq 11016

Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltexnq	⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnq

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 10967	. . . 4 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5749	. . 3 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3		ordpinq 10984	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
4		elpqn 10966	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ (N × N))
6		xp1st 8047	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐴) ∈ N)
8		elpqn 10966	. . . . . . . . 9 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ (N × N))
10		xp2nd 8048	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
11	9, 10	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐵) ∈ N)
12		mulclpi 10934	. . . . . . 7 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
13	7, 11, 12	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
14		xp1st 8047	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
15	9, 14	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐵) ∈ N)
16		xp2nd 8048	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
17	5, 16	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐴) ∈ N)
18		mulclpi 10934	. . . . . . 7 ⊢ (((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
19	15, 17, 18	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
20		ltexpi 10943	. . . . . 6 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N ∧ ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
21	13, 19, 20	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
22		relxp 5702	. . . . . . . . . . . 12 ⊢ Rel (N × N)
23	4	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 ∈ (N × N))
24		1st2nd 8065	. . . . . . . . . . . 12 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
25	22, 23, 24	sylancr 587	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25	oveq1d 7447	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
27	7	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐴) ∈ N)
28	17	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐴) ∈ N)
29		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝑦 ∈ N)
30		mulclpi 10934	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
31	17, 11, 30	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
32	31	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33		addpipq 10978	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ (𝑦 ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
34	27, 28, 29, 32, 33	syl22anc 838	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
35	26, 34	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
36		oveq2 7440	. . . . . . . . . . . 12 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
37		distrpi 10939	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦))
38		fvex 6918	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐴) ∈ V
39		fvex 6918	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝐴) ∈ V
40		fvex 6918	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐵) ∈ V
41		mulcompi 10937	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
42		mulasspi 10938	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
43	38, 39, 40, 41, 42	caov12 7662	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = ((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
44		mulcompi 10937	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N 𝑦) = (𝑦 ·N (2nd ‘𝐴))
45	43, 44	oveq12i 7444	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦)) = (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴)))
46	37, 45	eqtr2i 2765	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦))
47		mulasspi 10938	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵)))
48		mulcompi 10937	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))
49	48	oveq2i 7443	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵))) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
50	47, 49	eqtri 2764	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
51	36, 46, 50	3eqtr4g 2801	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)))
52		mulasspi 10938	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
53	52	eqcomi 2745	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)))
55	51, 54	opeq12d 4880	. . . . . . . . . 10 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩)
56	55	eqeq2d 2747	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ ↔ (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
57	35, 56	syl5ibcom 245	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
58		fveq2 6905	. . . . . . . . 9 ⊢ ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
59		adderpq 10997	. . . . . . . . . . 11 ⊢ (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
60		nqerid 10974	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
61	60	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐴) = 𝐴)
62	61	oveq1d 7447	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
63	59, 62	eqtr3id 2790	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
64		mulclpi 10934	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
65	17, 17, 64	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
67	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐵) ∈ N)
68	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐵) ∈ N)
69		mulcanenq 11001	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
70	66, 67, 68, 69	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
71	8	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 ∈ (N × N))
72		1st2nd 8065	. . . . . . . . . . . . . 14 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
73	22, 71, 72	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
74	70, 73	breqtrrd 5170	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵)
75		mulclpi 10934	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
76	66, 67, 75	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
77		mulclpi 10934	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (2nd ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
78	66, 68, 77	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
79	76, 78	opelxpd 5723	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N))
80		nqereq 10976	. . . . . . . . . . . . 13 ⊢ ((⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
81	79, 71, 80	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
82	74, 81	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵))
83		nqerid 10974	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Q → ([Q]‘𝐵) = 𝐵)
84	83	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐵) = 𝐵)
85	82, 84	eqtrd 2776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = 𝐵)
86	63, 85	eqeq12d 2752	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
87	58, 86	imbitrid 244	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
88	57, 87	syld 47	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
89		fvex 6918	. . . . . . . 8 ⊢ ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) ∈ V
90		oveq2 7440	. . . . . . . . 9 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → (𝐴 +Q 𝑥) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
91	90	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → ((𝐴 +Q 𝑥) = 𝐵 ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
92	89, 91	spcev 3605	. . . . . . 7 ⊢ ((𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
93	88, 92	syl6 35	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
94	93	rexlimdva 3154	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
95	21, 94	sylbid 240	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
96	3, 95	sylbid 240	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
97	2, 96	mpcom 38	. 2 ⊢ (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
98		eleq1 2828	. . . . . . 7 ⊢ ((𝐴 +Q 𝑥) = 𝐵 → ((𝐴 +Q 𝑥) ∈ Q ↔ 𝐵 ∈ Q))
99	98	biimparc 479	. . . . . 6 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) ∈ Q)
100		addnqf 10989	. . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q
101	100	fdmi 6746	. . . . . . 7 ⊢ dom +Q = (Q × Q)
102		0nnq 10965	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
103	101, 102	ndmovrcl 7620	. . . . . 6 ⊢ ((𝐴 +Q 𝑥) ∈ Q → (𝐴 ∈ Q ∧ 𝑥 ∈ Q))
104		ltaddnq 11015	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
105	99, 103, 104	3syl 18	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q (𝐴 +Q 𝑥))
106		simpr 484	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) = 𝐵)
107	105, 106	breqtrd 5168	. . . 4 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q 𝐵)
108	107	ex 412	. . 3 ⊢ (𝐵 ∈ Q → ((𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
109	108	exlimdv 1932	. 2 ⊢ (𝐵 ∈ Q → (∃𝑥(𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
110	97, 109	impbid2 226	1 ⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃wrex 3069  ⟨cop 4631   class class class wbr 5142   × cxp 5682  Rel wrel 5689  ‘cfv 6560  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  Ncnpi 10885   +_N cpli 10886   ·_N cmi 10887   <_N clti 10888   +_pQ cplpq 10889   ~_Q ceq 10892  Qcnq 10893  [Q]cerq 10895   +_Q cplq 10896   <_Q cltq 10899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8746  df-ni 10913  df-pli 10914  df-mi 10915  df-lti 10916  df-plpq 10949  df-mpq 10950  df-ltpq 10951  df-enq 10952  df-nq 10953  df-erq 10954  df-plq 10955  df-mq 10956  df-1nq 10957  df-ltnq 10959

This theorem is referenced by:  ltbtwnnq  11019  prnmadd  11038  ltexprlem4  11080  ltexprlem7  11083  prlem936  11088